Definition of a sphere
A sphere in \(\mathbb{R}^3\) is defined by point vector \(\mathbf{c} \in \mathbb{R}^3\) and a scalar \(r \in \mathbb{R}, r \geq 0\). \(\mathbf{c}\) is called the center of the sphere, \(r\) is called the radius of the sphere. The sphere is the set of all point with a distance to \(\mathbf{c}\) smaller than or equal to the radius \(r\) i.e. \( \left\{ \mathbf{p} \in \mathbb{R}^3 \;|\; |\mathbf{p} - \mathbf{c}| \leq r \right\} \).